Generic central extensions and projective representations of finite groups

Any free presentation for the finite group $G$ determines a central extension $(R,F)$ for $G$ having the projective lifting property for $G$ over any field $k$. The irreducible representations of $F$ which arise as lifts of irreducible projective representations of $G$ are investigated by considering the structure of the group algebra $kF$, which is greatly influenced by the fact that the set of torsion elements of $F$ is equal to its commutator subgroup and, in particular, is finite. A correspondence between projective equivalence classes of absolutely irreducible projective representations of $G$ and $F$-orbits of absolutely irreducible characters of $F'$ is established and employed in a discussion of realizability of projective representations over small fields.